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    On the maximum quartet distance between phylogenetic trees

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    A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on nn leaves is at most (23+o(1))(n4)(\frac 23 +o(1))\binom{n}{4}. Using the machinery of flag algebras we improve the currently known bounds regarding this conjecture, in particular we show that the maximum is at most (0.69+o(1))(n4)(0.69 +o(1))\binom{n}{4}. We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most (23+o(1))(n4)(\frac 23 +o(1))\binom{n}{4}.Comment: arXiv admin note: text overlap with arXiv:1203.272
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